. '

Nr. 175/94

GENERALIZED FUNCTIONALS IN GAUSSIAN SPACES

- THE CHARACTERIZATION THEOREM REVISTED -

Yu. G. Kondratiev, P. Leukert, J. Pott hoff,L. Streit, W. Westerkamp .

February 1994

GENERALIZED FUNCTIONALS IN GAUSSIAN SPACESt

- THE CHARACTERIZATIONTHEOREMREVISITED-

Yu.G. Kondratie'li, P. LeukertI, J. Potthoif, L. Streit1,2, W. Westerkampl

1 BIBoS, Universität Bielefeld2 Centro das Cieneias Matematieas, Universidade da Madeira3 Lehrstuhl für Mathematik V, Universität Mannheim

Abstract. Gel'fand tripies of test and generalized functionals in Gaussian spaees areeonstructed and eharacterized.

1. Introduction

In reeent years there was an inereasing interest in white noise analysis, due to its rapiddevelopments in mathematieal strueture and applieations in various domains. Especially,the eircle of ideas going under the heading 'eharaeterization theorems' has played quite animportant role in the last few years. These results [21], [33], [42], and their variations andrefinements (see, e.g., [31], [34], [37]' [48], [50], [54], and referenees quoted there), providea deep insight into the structure of spaees of smooth and generalized random variablesover the white noise spaee or - more generally - Gaussian spaces. Also, they allow forrat her straightforward applications of these notions to a number of fields: for example,Feynman integration [11], [16], [20], [32], representation of quantum field theory [2], [43],stochastie equations [7], [30], [39], [40], [41], interseetion loeal times[10], [49], Diriehletforms [3]' [4], [15], infinite dimensional harmonie analysis [14] and so forth. Moreover,eharaeterization theorems have been at the basis of new methods for the construction ofsmooth and generalized random variables [24], [34]whieh seem to be useful in applieationsuntraetable by existing methods (e.g., [18], [19]).

The purpose of the present article is four-fould: We wish 1. to clarify and generalizethe structure of the existing characterization theorems, and at the same time, 2. to reviewand unify reeent developments in this direction, 3. to establish the eonneetion to rieh,related mathematiealliterature [1], [8]' [12], [45], [53], whieh might be helpful in futuredevelopments, and - last but not least - 4. to fill a gap in the article [42]. In this sense,

tSupported by program STRIDE.

1

the present paper attempts to give known results and some existing 'folklore' around thema general form that can be used as a reference for future research. In the course of doingthis, we also establish some new results, for instance an analytic extension property of U-functionals, and the topological invariance of certain spaces of generalized random variableswith respect to different construction schemes.

This article is organized as follows. In Section 2 we present some notions and resultsfrom complex analysis ontopological vector spaces, from Gaussian analysis, and from Fockspace theory. In particular, we construct a nuclear rigging

(Af) c f(H) c (Af)*

of the symmetrie Fock space f(1-l) over a Hilbert space H. We give a construction ofthe second quantized space (Af) solely in terms of the topology of Af, independent of theparticular representation as a projective limit. Via the well-known Wiener-1t6-Segal iso-morphism this provides a rigging of the L2-space over a Gaussian measure space by spacesof' smooth and generalized random variables. In Section 3 we study the U-functionalsassociated with the elements in (Af) and (Af)*. We derive an analytic extension propertyfor U-functionals, and use this to prove theorems which characterize (Af) and (Af)* interms of their S-transforms. In Seetion 4 we prove two corollaries of the characterizationtheorem for (Af)* which appear to be useful in applications.

2.Preliminaries

2...1 G-Entire FunetionsWe provide some well-known facts from complex analysis on topological vector spaces (see,e.g., [8] and [12]) with a view towards applications in the next section.

Let f be a locally convex complex vector space,

Definition 1. A mapping P from f into ([7is ealled an n-homogeneous polynomial,if it is the eomposition of the diagonal mapping ~n : x I--t (x, x, ... , x) from f into fn andasymmetrie n-linear mapping L from fn into ([7, i.e., P = L 0 ~n = L. Let Pn(£) denotethe spaee of all n-homogeneous polynomials.

Definition 2. A funetion F dejined on £ with values in ([7is saidto be G-entire if forall e,7]E £ the eomplexvalued funetion

Z I--t F( 7] + ze),z E ([7,

is entire. Let HG(f) denote the set of all G-entire mappings from £ into ([7 .

. For F E HG(£), 7] E £, there exists a unique sequence (~dn-W7]), nEIN 0) of homo-,. ," ' --.. n.

geneous polynomials ~! dn F( 7]) E Pn(£), nEIN, such that for all e E £,

'..

(1)

2

-Of course, dn F( 11)( 0 is the n-th partial derivative of F at 11 in the direetion e. Thecorresponding n-linear form is denoted by dnF( 11)( 6,... ,en), 6, ... ,en E £.Definition 3. Let F be a mapping /rom £ into (V. F is called entire if it is in Ha(£),and if it is continuous. H(£) denotes the space of all entire funetions on £.

Proposition 4. Let F EHa(£). Then F EH(£) if and only if F is locally bounded.

We conclude this subsection by stating a result which is related to the celebrated"cross theorem" of Bernstein. For a review of such results we refer the interested readeralso to [1]. The followiIig is a special case of a resultby Siciak: if we make use of thefact that any segment of the realline in the complex plane has strictly positive transfinitediameter, then Corollary 7.3 in [45] implies

Proposition 5. Let nEIN, n ~ 2, and f be a complex valued funetion on IRn• Assumethatfor all k = 1,2, ... ,n, and (XI, ... ,Xk-I,Xk+I,'",xn) E IRn-I, the mapping

from IR into (V has an entire extension. Then f has an entire extension to (Vn.

2.2 Gaussian SpacesThe primordial object of Gaussian analysis (e.g., [6]' [17], [21], [22], [25], [26], [27], [28],[29]) is areal separable Hilbert space 1i. One then considers a rigging of 1i,N c 1i c N*,where N is areal nuclear space (see below and [13]), densely and continuously embeddedinto 1i, and N* is its dual (1i being identified with its dual). A typical example (whichappears for instance in white noise analysis) is the rigging S(IR) c L2(JR) c S'(IR) ofL2(IR) (with Lebesgue measure) by the Schwartz spaces of test funetions and tempereddistributions.

Via Minios' theorem the canonical Gaussian measure Jl on N* is introduced by givingits characteristic funetion

CU) = L. ei(w,f} d/l(w) = e-~Ifl~, fE N.

The space L2 (N* ,d/l) = (L2) of (equivalence classes of) complex valued funetions on N*which are square-integrable with respeet to /l has the well-known Wiener-Itc3-Segal chaosdecomposition [36]' [46], [47]' and one has the familiar Segal isomorphism I between (L2)and the com'plex Fock space r(1i) over the complexification 11.c of 1i.

Spaces of smooth funetions on N* canbe construeted by mapping appropriate sub-spaces of r(1i) into (L2) via the unitary mapping I-I: r(1i) -+(L2), see, e.g., the con-struction using second quantized operators in [6], [17]. In the present context, we prefer towork exclusively in Fock space. At all times a 'translation' into funetion space languagevia I-I is of course equally valid.

3

2.3 Second Quantized SpacesDur starting pointis areal separable nuclear spaee N. It is well-known (e.g., [38]) thatthe topology ofN is equivalent to the projeetive limit topology of an inereasing eountablesystem (I . Ip, pEIN 0) of compatible Hilbertian norms I . Ip, In other words, N is aeountably Hilbert spaee [13]"

N == n'Hp,p

wh,ere 'Hp is equal to the eompletion of N with respeet to I . Ip, Moreover, the usuallIilbert-Schmidtproperty for the embeddings holds, i.e., for every pEIN 0 there exists ap' > p so thatthe embedding f., from 'Hp' into Hp is a Hilbert-Sehmidt operator. We shalldellote the bilinear dual pairing on N* x N by (., .). A special role - also with a viewto~ards Gaussian spaees - is played by the Hilbert spaee 'Ho, whieh we also denote by 'H.

Consider the Fock space, e.g., [9]' [46]' over 'H00

r ('H) = E9 'H~n,n=O

(2)

~in' 1{@nWn e,p

with respeet to the inner produet determined by

(@n ol.@n) 2nq '( ol.)n'P , 'f/ rq(1ip) = n. 'P, 'f/ 1ic,p'

and denote the eorresponding norms by 11.11 . Finally we set, p,q

~where 'H~n is the symmetrie n-fold tensor produet of'He with itself. The inner produetand norm 11.110 .of f('H) are generated by

('P@n,l/>@n)r(1i) = n!('P,l/»~c'

Likewise" for pEIN 0, nEIN, 'H~,; denotes the n-fold symmetrie tensor produet of 'He,pwith itself, and it is eonsfdered as a subspace of 'H~,;.The eanonieal norm of the latteris denoted by I . Ip, too (the meaning will be clear from the eontext). The duals of 'H~n. ..-. ..-. - ,pand 'H~,;, respeetively, aredenoted by 'H~,~p and 'H~,~p, respeetively. The Hilbertiannormof'H~,~p is denoted by I . I-p, and we remark that for nEIN, p E 7Z, <p(n) E 'H~,;,lqi(n) Ip is equal to the Hilbert-Sehmidt norm of <p(n) considered as a linear form on 1i~,~p'For q E IN 0, we introduee Hilbert spaees rq (Hp) as the eompletions of the spaee of finitedireet sums

p,q

equipped with the projeetive limit topology.

Remarks. Evidently substitution of the value 2 in equation (2) by any other numberstrietly larger than 1 produces the same spaee (N). We use the same notation (N) for thenuclear subspace of (L2) eorresponding to (N) under the Wiener-Ito-Segal isomorphism.

4

Lemma 6. (N) is nuclear.

ProoE. Nuclearity of (N) follows essentially from that of N. For fixed p, q consider theembedding

I: rq, ('Hp') --+ rq ('Hp)where p' is chosen such that the embedding

is Hilbert-Schmidt. Then

n

Hs Hilbert-Schmidt norm is easily estimated by using an orthonormal basis (cf., e.g., [17],Appendix A.2) of rq,('Hp')' The result is the bound

00

IIIII~s :::;L 2n(q-q') II"II~~n=O

which is finite for suitably chosen q'. o

Theorem 7. The topology on (N) is uniquely determined by the topology on N.

ProoE. Let us assurne that we are given two different systems of Hilbertian norms I. Ipand I'I~ , such that they induce the same topology on N . For fixed k and I we haveto estimate lI'II~,1 by 1I'lIp,q for some p, q (and vice versa which is completely analogous).Sillce I.I~has to be continuous with respeet to the projective limit topology on N, thereexists p and a constant C such that Ifl~ :::;C Iflp' for all f E N, i.e., the injection " from'Hp into the completion Kk of N with respeet to I . I~is a mapping bounded by C. Wedenote by " also its linear extension from 14J,p into KC,k. It follows from a straightforwardmodification of the proof of the Proposition on p. 299 in [44], that ,,@n is bounded by cn

@ @ ~from 'Hc,; into Kc,~' Now we choose q such that 2 2 ~ C. Then

00

lI'II~,1 = Ln! 2n1 I'I~n=O00

:::;Ln! 2n1c2n 1'1;n=O

SII'II;,q'which had to be proved. 0

From general duality theory on nuclear spaces we kllOWthat the dual of (N) is givenby

p,q

5

anel one verifies thatf q (Hp)* = f _q (H-p).

We shall denote the bilinear dual pairing on (N)* x (N) by ((., .)) :

00

((<f?,'P)) = Ln! (<f?(n), 'P(n)),n=O

where <f?~ f_q(H-p) corresponds to the sequence (<f?(n), nE INo) with <f?(O)E C, and

<f?(n)E H2,~p' nEIN.

Remark. Consider the particular choice N = S(lR). Then (N)(*) coincide with (the Fockspace equivalents of) the well-known spaces (S)(*) of white noise funetionals, see, e.g.,[17],[42]. For the norms 1I'Pllp = IIf(AP)'Pllo introduced there, we have 11 . IIp = 11 . IIp,o, anel11 . IIp,q ~ 11 . IIp+f' More generally, if the norms on N satisfy the additional assumptionthat for all p 2: 0 and all c > 0 there exists p' 2: 0 such that I . Ip ~ cl . Ip', then thecOllstruction of Kubo and Takenaka [26] (and other authors) leads to the same space (N).The construction presented here has the advantage of being manifestly independent of thechoice of any concrete system of Hilbertian norms topologizing N.

For the exponential veetors

00 1<PI := 2.: ,. t~n

n.n=O

one calculates the norms1I<p fll;,q = e2Qlfl;,

anel hence for all f EN they are in (N). This then allows for the following

Definition 8. Lei <f?E (N)*. The S-transform of <f?is ihe mapping from N inio Cgiven by

S<f?(J) := ((4!,<PI)), fE N.

We note that the exponential vectors {<p f, f E N}, are a total set £ in (N), and henceelements of (N)* are characterized by their S-transforms. Furthermore, it is obvious thatthe S-transform.of <f?E (N)* extends to Ne: for e E Ne set S<f?(e)= ((<f?,<Pe)), where <Peis the complex exponential veetor l:n ~!erzln E (N).

3. U-Funetionals and the Charaeterization Theorems

We begin with adefinition.

6

Definition 9. Let F : N ---+ <Cbe such that

C.1 for all f, gEN, the mapping A 1---+ F(g + Ai) from IR into <C has an entireextension to A E <C,

C.2 for some continuous quadratic form B on N there exists constants C, K > 0 suchthat for all f E N, z E <C,

IF(zi)1 ~ C exp(K IzI2IB(f)I).

Then F is called a U-functiona1.

Remark. Condition C.2 is actually equivalent to the more conventional

C.2' there exists constants C, K > 0 and pEIN 0, sothat for all f E N, z E <C,

(3)

Lemma 10. Every U -funetional F has a unique extension to an entire funetion on Ne.Moreover, if the bound on F holds in the form (3) then for all pE (0,1),

with C' = C(1 - p)-~, K' = 2p-1e2 K.

ProoE. First we show that a U -funetional F has a,G-entire extension. The extension of F(denoted by the same symbol) is given by F(T]) = F(go +zgd, TJ = go +zgl E Ne, gO,91 EN, z E <C. Let e E Ne be of the form e = g2 + ig3, 92,93 E N. Consider the mapping

from IR3 into <C. Condition C.l and Proposition 5 imply that this funetion has an entireextension to <C3• In particular, Fis G-entire on Ne.

Let e E Ne, and consider the Taylor expansion of F( 0 at the origin (cf. (1»:

(4)

For all f E N, nEIN, R> 0, we obtain from C.2' and Cauchy's inequality the estimate

We choose R = (2~()~' and get for f EN with Iflp = 1 the inequality

-- (2eK)n/2IdnF(O)(f)1 ~ Cn! 7 .

7

A standard polarization argument (see, e.g.,(35, g3]) andhomogeneity of <in F(O) yield thefollowing bound for the n-linear form dn F(O):

n

Idnp(O)(fl, ... ,fn)l::; C(n!(2e2Kr)~ IIIfklp,k=l

where ft, ... , in E N (and we used :~ ~ en).

Since dn F(O) is n-linear on Ne, the last inequality gives the estimate

n

Idn F(O)(~l, ... , ~n)l ~ C (n! (4e2 Kr) ~TI I~klp,. k=l

(5)

(6)

.

far 6, ... ,en E Ne. In particular, the Taylor coefficients in (4) have absolute valuebounded by

and we get (by Schwarz' inequality) the following estimate for all p E (0,1),

Hcnce Fis locally bounded on Ne, and therefore Proposition 4 implies that Fis entire. 0

Now we are ready to prove the following generalization of the main result in [42] whichcharaeterizes the space (N)* in terms of its S-transform.

Theorem 11. A mapping F :N -+ (D is the S -transform of an element in (N)* if andonly if it is a U-funetional.

Proof. Let ,cI»E (N)*. Then cI»E r_q(1-l_p) for some p, q E IN o. As we have remarkedat the end of Section 2, the S-transform of cI»extends to Ne, and therefore it makes senseto consider the mapping ~ f-+ ScI»(~)from Ne into (D. We shall show that this mapping isentire. We have

00

ScI»(O= I:(cI»(n),e@n), e E Ne.n=O

We estimate as follows:

00

IScI»(OI~ L 1~(n)l_pl~l;n=O

8

The last estimation shows that the power series for Sep on Ne converges uniformlyon everybounded neighborhood of zero in Ne, and therefore it defines an entire function on thisspace [12]. In particular, C.1 holds for Sep. Moreover, the choice e = zj, z E ([;,. JEN,shows that also C.2' is fulfilled. Hence Sep is a U-functional.

Conversely let F be a U-functio,nal. We may assurne the bound in the form (3).Consider the n-linear form dn F(O) on Ne constructed in the proof of Lemma 10. Theestimate (6) shows that dn F(O) is separately continuous on Ne in its n variables. Henceby the nuclear theorem (e.g., [6], [13]) there exists ep(n) E (N;;)~n so that

6, ... ,en E Ne.

Let p' > p be such that the embedding t : 'Hp' --+ 'Hp is Hilbert-Schmidt, and let(ek, kEIN) be an orthonormal basis of'Hp' in N. For nEIN, (ek1 0 ... 0 ekn, ki EIN, i = 1, ... ,n) is an orthonormal basis of 'H~,;" Then we can estimate in the followingway (cf. (5»:

lep(n)l:.p' = L l(ep(n),ek1 0 ... 0 ekn)12

k1, ••• ,kn

k1, ••• ,kn. 00

~ C2(n!)-1(2e2 Kt(L: Itekl;tk=l

= C2(n!)-1(2e:lJ( IItll~st...•..

• .=r..(n) E '1JiZ>n dl.e., ":i:' , t.e,_p" an

lIep(n)lI:'p',_q ~ C2(21-qe2 K IItll~st.For cP given by the sequence (ep(n), nE INo) (ep(O) = F(O» we have

00

1I~II:'p',_q::;C2 I)21-qe2 K IItll~s)nn=O

= C2(1 - 21-qe2 K IItll~s)-l< +00,

(7)

if we choose q large enough so that 21-qe2 K IItll~s < 1. In particular, ep E (N)*, and forfE N we have by (4),

00

Sep(f) = L (ep(n), fiZ>n)n=O

= f ~dnF(O)(f)n.n=O= F(f).

9

Uniqueness of <P:- S-lF follows from the fact that the exponential vectors are total in(N). . . 0

As. a by-proeluet of the above proof we obtaiIi the following localization result forgeneralizeel functionals.

Gorollary 12. Given a U-junctional F satisfying C.2'. Let p' > p be such that theembedding.,,: 1-lp'~ Hp is Hilbert-Schmidt, and q E lNo so that p:= 21-qe2J{ 11"II~s < 1.Then <P:= S-l FE r-q(H-p')' and

1I<pII_p',_q ~ C(I- p)-1/2. (8)

For analogous results in white noise analysis see, e.g., [23]' [37], [50].

..Withinthe framework establisheel here one can treat the following anel numerous otherexamples in a unifiedway.

Example 13. We choose the triplet

anel equip S'(JRn) with the Gaussian measure with characteristic functional

Then the framework allows to eliscuss functionals of white noise with n-dimensional timeparameter [48].

Example 14. If we choose a finite direct sum of identical copies of S(JR) as the basicreal nuelear space we obtain the characterization of the space of Hida distributions of thenoise of an n-dimensional Brownian motion [48].

We elose this section by the corresponding characterization theorem for (N). Thisresult is independently due to [21), [31), [33), and has been generalized and modified invarious ways, e.g., [37], [50], [54].

Theorem 15. A mapping F :N ~ (J7 is the S -transform of an element in (N) if andonly if it admits C.l and the following condition

C.9 there exists a system of norms (/"_P' pEIN 0), which yields the inductive limittopology on N*, and such that for all p ~ 0 and f > 0 there exi,qts Cp,e > 0 sothat

(9)

If for F conditions C.l and C.3 are satisfied we say that Fis of order 2 and minimaltype.

10::

Proo£ If cp E (N) then condition C.l is satisfied as a consequence of Theorem 11. Forany p, q ~ 0 we estimate as follows

'. 00

IScp(zf)1 = IL (cp(n), (zf)0n) In=O00::;L Izlnlcp(n)lp Ifl~pn=O

Hence condition C.3, too, is necessary.

Conversely, let F be a U-fundional of order 2 and minimal type. From F, constructa sequence cp = (cp(n), nEIN 0) of continuous linear forms cp(n) on N@n as in the proof ofTheorem 10. We have to show that cp belongs to rq(Hr) for all T, q E IN o. Let T, q E IN 0

be given. Choose p > 1" such that the injection ,,: 7-lp --+ 7-lr is Hilbert-Schmidt.Then sois the injection ,,* : 7-l-r --+ 7-l_p• c > 0 in (9) is chosen so that p := c2l+qe211"*II~s < 1.Then the analogue of (7) reads

and we get00

IIcpllr,q = (L IIcp(n)II;,q)~n=O

/

_1

::; Cp,~(l - p) 2.

Thus cp E (N), and the proof is complete.

4. Corollaries

o

One useful application of Theorem 11 is the discussion of convergence of a sequence ofgeneralized fundionals. A first version of this theorem is worked out in [42]. Here we useour more general setting to state

Theorem 16. Let (Fn, n E .DV) denote a sequence of U -functionals such that

1. (Fn(f), nE .DV) is a Cauchy sequence for all f E.N,2. theTe exists a continuous norm 1.1 on N and C, K > 0 such that IFn(zf)1 <

CeKlzl2lf\2 for all fE N, z E lD, and for almost all nEIN.

Then (S-l Fn, nEIN) converges strongly in (N)*.

11

Proof The assumptions and inequality (8) imply that there exist p, q ~ Oand p E (0,1)such that for all nEIN,

where «Pn = 5-1 Fn. Since £ is total in r_q(H-p), assumption 1 implies that (((<<pn, 'P)), 11, EIN) is a Cauchy sequence for an'P E (N). Since (N)* is the dual of the countably Hilbertspace (N), which is in particular Frechet, it follows from the Banach-Steinhaus theoremthat (N)* is weakly sequentially complete. Thus there exists «PE (N)* such that «Pis theweak limit of (cPn, n E lfV). The proof is concluded .by the remark that weak and strongconvergence of sequences coincide in the duals of nuclear spaces (e.g., [13]). 0

As a second application we consider a theorem which concerns the integration of afamily of generalized functionals.

Theorem 17. Let (A, A, v) be a measure space, and ). ~ «PAa mapping from A to (N) * .We assume that the 5 -transform FA = 5«pA satisfies the following conditions:

1. for every f E N the mapping ). ~ FA (I) is measurable,2. there exists a continuous norm I . I on N so that forall ). E A, FA satisfies the b01tnd

IFA(zf)' ..:::; CAeK>.lzI2IfI2, and such that). ~ KA is bounded v-a.e., and ). ~ CA isintegrable with respeet to v.

Then there are q,p ~ 0 such that cI>. is Bochner integrable on r_q (H_p). Thus in partic-ular, I «PAdv().) E (N)* ,

and

Proof In inequality (3) for FA(zf) we can replace KA by its bound. With this modifiedestimate and Corollary 12 we can find p,q ~ 0 and p E (0,1) such that for an). E A,

(10)

Since the right hand side of (10) is integrable with respect to v, we only need to showthe weak measurability of). ~ «PA(see [52]). But this is obvious because ). ~ ((«pA,'P))ismeasurable for an 'P E £ which is total in r q(Hp). 0

Acknöwledgement. We thank Professors S. W. He and H. Sato for pointing out thegap in [42]' and Professor B. 0ksendal for helpful discussions. We owe special thanks toProfessor L.I. Ronkin who taught us about Bernstein's theorem and its generalizations.

12

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